In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
General definition
Given a collection S of sets, consider the Cartesian product X = \prod_{Y\in S} Y of all sets in the collection. The canonical projection corresponding to some Y\in S is the function p_{Y} : X \to Y that maps every element of the product to its Y component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form,
\bigcap_{i=1}^n p_{Y_i}^{-1} \left(A_i\right) = \left\{ \left(x\right) \in X \mid p_{Y_1}(x) \in A_1, \dots, p_{Y_n}(x) \in A_n\right\}
for any choice of n, finite sequence of sets Y_1,...Y_n\in S and subsets A_{i} \subseteq Y_i for 1 \leq i \leq n.
Then, when all sets in S are topological spaces, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form \bigcap_{i=1}^n p_{Y_i}^{-1} \left(U_i\right) where for each i, U_i is open in Y_i. In the same manner, in case of measurable spaces, the cylinder σ-algebra is the one which is generated by cylinder sets corresponding to the components' measurable sets.
The restriction that the cylinder set be the intersection of a finite number of open cylinders is important; allowing infinite intersections generally results in a finer topology. In the latter case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.
Cylinder sets in products of discrete sets
Let S = \{1,2,\ldots,n\} be a finite set, containing n objects or letters. The collection of all bi-infinite strings in these letters is denoted by
S^\mathbb{Z} = \{ x = (\ldots, x_{-1}, x_0, x_1, \ldots) :
x_k \in S \; \forall k \in \mathbb{Z}\}.
The natural topology on S is the discrete topology. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on S^\mathbb{Z} are
C_t[a]= \{x \in S^\mathbb{Z} : x_t = a \}.
The intersections of a finite number of open cylinders are the cylinder sets
\begin{align}
C_t[a_0, \ldots, a_m] & = C_t[a_0] \,\cap\, C_{t+1}[a_1] \,\cap \cdots \cap\, C_{t+m}[a_m] \\
& = \{x \in S^\mathbb{Z} : x_t = a_0, \ldots ,x_{t+m} = a_m \}
\end{align}.
Cylinder sets are clopen sets. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but the complement of a cylinder set is a union of cylinders, and so cylinder sets are also closed, and are thus clopen.
Definition for vector spaces
Given a finite or infinite-dimensional vector space V over a field K (such as the real or complex numbers), the cylinder sets may be defined as
C_A[f_1, \ldots, f_n] = \{x \in V : (f_1(x),f_2(x),\ldots,f_n(x)) \in A \}
where A \subset K^n is a Borel set in K^n, and each f_j is a linear functional on V; that is, f_j\in (V^*)^{\otimes n}, the algebraic dual space to V. When dealing with topological vector spaces, the definition is made instead for elements f_j \in (V')^{\otimes n}, the continuous dual space. That is, the functionals f_j are taken to be continuous linear functionals.
Applications
Cylinder sets are often used to define a topology on sets that are subsets of S^\mathbb{Z} and occur frequently in the study of symbolic dynamics; see, for example, subshift of finite type. Cylinder sets are often used to define a measure, using the Kolmogorov extension theorem; for example, the measure of a cylinder set of length m might be given by 1/m or by 1/2m.
Cylinder sets may be used to define a metric on the space: for example, one says that two strings are ε-close if a fraction 1−ε of the letters in the strings match.
Since strings in S^\mathbb{Z} can be considered to be p-adic numbers, some of the theory of p-adic numbers can be applied to cylinder sets, and in particular, the definition of p-adic measures and p-adic metrics apply to cylinder sets. These types of measure spaces appear in the theory of dynamical systems and are called nonsingular odometers. A generalization of these systems is the Markov odometer.
Cylinder sets over topological vector spaces are the core ingredient in the[citation needed] definition of abstract Wiener spaces, which provide the formal definition of the Feynman path integral or functional integral of quantum field theory, and the partition function of statistical mechanics.